Understanding Projections

Goal

Understand details about spatial data to gather how a point on a 3-d sphere can be presented in a two-dimensional plane.

Outline

- Datums
- Units
- What is a projection?
- Globe demonstration 
- Different types of projections - D3 example
- Why do we care?

Understanding GIS Data

Question:

How could we tell someone on the other side of the world where we want to meet them for coffee?

It depends on how we talk about the world.

• Do we use the same units?
• Do we understand direction the same?

Thankfully, we can communicate mathematically.

So About the Earth

Well it's not flat.

Earth's Geometry

The earth actually isn't even a perfect sphere,
especially given the topographic changes of mountain ranges and valleys.

Earth Elevation Profile

Sphereoid

We call the shape of the Earth a spheroid.

A sphereoid is defined by:

• Semi-Major Axis
• Semi-Minor Axis

Spheroids

And there are many approximations (meters):

• Sphere; 6370997; 6370997
• Clarke (1866); 6378206.4; 6356583.8
• GRS80 (1980); 6378137; 6356768.33
• WGS1984 (1984); 6378137; 6356752.31

Datum

A datum is built on top of the selected spheroid and can incorporate local variations in elevation.

Basically, we are setting the origin of our sphere as if it were placed on the earths actual geoid.

For example: Bellingham, WA

Datum	Longitude	Latitude
NAD 1927	-122.46690368652	48.7440490722656
NAD 1983	-122.46818353793	48.7438798543649
WGS 1984	-122.46818353793	48.7438798534299

• NAD27 is quite different becasue it uses different spheroid and daturm
• NAD83 and WGS84 express coordinates that are nearly identical
  • same spheroid
  • different dataum

Spherical Measurements

So what do we know about Circles/Spheres:

• 360 degrees

Break down a sphere into:

• Latitute; parallels
• Longitude; meridians

Origins on the Earth:

• Prime Meridian
• Equator

Together form: Graticule

Measurements on a Sphere

Both of these are measurements from the Earth's center to a point of the Earth's surface.

These two allow us to find any place on Earth.

Cool!!! And it's called:

Geoographic Coordinate System

What is this actually measuring...?

• Degrees

Units:

• Decimal Degrees
• Sexagesimal Degree: Degrees, Minutes, Seconds

Unit Equivalencies

$$1° = 60' = 3600"$$

$$1' = (\frac{1}{60})° = 0.01666667°$$
$$1" = (\frac{1}{3600})° = 2.77778e^{-4}° = 0.000277778°$$

Well what does that mean...?
Described as a Coordinate Pair: (x,y) <- (in most cases)

University of Puget Sound:

Decimal Degrees:

• Latitude: 47.261780
• Longitude: -122.481689
• (-122.481689, 47.261780)

Sexagesimal

• 47 Degrees 15 Minutes 42.4080 Seconds
• 122 Degrees 28 Minutes 54.0804 Seconds

Conversion

DM -> DD:

$$decimal degrees = degrees + \frac{Minutes}{60} + \frac{seconds}{3600}$$

DD -> DM:

$$absdd = |dd|$$

$$degrees = integer(absdd)$$ $$minutes = integer((absdd - degrees) * 60)$$
$$seconds = ((absdd - degrees - \frac{minutes}{60}) * 3600)$$

Measurements on a Sphere Continued

1 degree of longitude at the equator: 111.321 KM

1 degree of longitude at 60 degrees of latitue: 55.802 KM

Woah, thats weird...

Ends up, most people are not good at thinking in 3 dimensional space...

What are our Alternatives?

Convert to Planar Coordinate System!

• Convert a 3d sphere to a 2d plane.

Unlike Geographic Coordinate Systems, length, angle, and area are contstant in both the x and y directions.

University of Puget Sound:

UTM Zone 10 (FT)

• Easting:
• Northing: 5234383.60405841
• (5234383.60405841, 539230.069260529)

WA State Plane South (FT):

• Latiutude: 1148411.465
• Longitude: 709443.892
• (709443.892, 1148411.465)

This is Called a Projection

We can use different types of math here to be able to flatten the sphere to a 2d plane.

Different projection formulas result in different types of projections.

Additionally, different projection formulas can be used to preserve angles, area, or distance.

The four spatial properties that are subject to distortion are: shape, area, distance and direction

Projection Properties

Conformal Projections

• Preserve local shape
• If graticules are intersecting at 90 degrees, can also preserve local angles

The area enclosed by intersection of arcs may be distorted.

Equal-Area Projections

• Preserves area

Shape, angle, scale may all be distorted. Graticules may not intersect at right angles.

Eqidistant Projections

• Preserves distance

Scale is only preserved in some areas of a map.

Azimuthal Projections (True-Direction Projections)

• Preserves direction

Each map projection is good at preserving only one or two of the four spatial properties.

So when working with small-scale (large area) maps and when multiple spatial properties are to be preserved, it is best to break the analyses across different projections to minimize errors associated with spatial distortion.

Geodesic geometries

Projected coordinate systems will introduce errors in measurements.

This is due to the difference in computing the distance between two points. On a sphere, the distance traveled is in an arc, but in a planar coordinate system, typically a linear system.

Distance between two points in a planar system:

Point(4, 1)
Point(5, 6)

$$distance = \sqrt{(x_2 - x_1)^{2} + (y_2 - y_1)^{2}}$$

A geodesic distance is the shortest distance between two points on an ellipsoid (or spheroid).

A geodesic area measurement is one that is measured on an ellipsoid.

Most of the time, the scales we are producing maps at allow us to ignore these errors, as the levels of tolerance in our calculations allow for such precision.

However, if we use small scale analysis (covering a large area of the earth), then often these levels of precision are exceeded.

Different Types of Projections

Albers Equal Area

Two standard parallels

Commonly used for maps of the conterminous United States.

• Conic
• Equal-area

• Shapes, directions, angles, and distances are generally distorted.
• The scale, directions, and distances are true only along the standard parallels

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Lambert Conformal Conic

Directions, angles, and shapes are maintained at infinitesimal scale.

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Equirectangular

The meridians and parallels are equally spaced straight lines forming a Cartesian grid.

Each rectangular grid cell has the same size, shape, and area only in the projected space.

• North, south, east, and west directions are always accurate.
• General directions are distorted, except locally along the standard parallels.

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Orthographic

• Neither conformal nor equal-area
• Shapes, areas, distances, directions, and angles are all generally distorted.
• Only the center of the map is free of distortion.

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Transverse Mercator

Used for mapping large-scale or smaller areas with predominantly north-south trending extents.

Conformal map projection

• Preseves Distance and Angles
  • Can use the Pythagorean theorem to be able to calculate distances
  • Great for in the field

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Universal Transverse Mercator

The UTM system divides the Earth into 60 zones, each 6° of longitude in width.

• Conformal Projection

UTM Zone 10 - Washington State

UTM Zone	Central Meridian	UTM Zone Extent
10	123°W	126°W-120°W

• Preserves distance in North-South Direction

State Plane

State Plane system of large-scale conformal map projections originally created in the 1930s to support surveying, engineering, and mapping activities throughout the U.S. and it's territories.

Loosely based on:

• Transverse Mercator projection
• Lambert conformal conic projection

Problems - Puget Sound Metro

King County: "Washington State Plane North"
Pierce County: "Washington State Plane South"

I have even encountered a custom projection to deal with this issue!!

Takeaways

There are a variety of different projections for different reasons and I don't expect you to know them all.

However, you will need to know how to find a projection of a data source, map, and other services to be able to work with data.

Each agency typically has their own standards and you will be asked to use that when working with that data.

Resources

Projections in ArcGISPro

Reading

Learning ArcGIS Pro 2: Chapter 3